A categorical version of the Lefschets-Nöbeling-Pontryagin theorem on embedding compacta in R n

Abstract

For a category K we use Ob(K) to denote the class of all objects of K; if X, Y ϵ Ob( K), then Mor K ( X, Y) is the set of all K-morphisms from X into Y. Let A and B be subcategories of the category of all topological spaces and their continuous maps. We say that a covariant functor F: A → B is an embedding functor if there exists a class { i x : X ϵ Ob( A)} satisfying the following conditions: 1. (i) i x : X → F( X) is a homeomorphic embedding for every X ϵ Ob( A), and 2. (ii) if X, Y ϵ Ob( A) and f ϵ Mor K ( X, Y), then F( f) o i x = i y o f. For a natural number n let C( n) denote the category of all n-dimensional compact metric spaces and their continuous maps. Let G(< ∞) be the category of all Hausdorff finite-dimensional topological groups and their continuous group homomorphisms. We prove that there is no embedding covariant functor F: C(1) → G(< ∞), but there exists a covariant embedding functor F: C(0) → G(0), where G(0) is the category consisting of the single (zero-dimensional) compact metric group Z 2 ω and all its continuous group homomorphisms into itself, i.e., Ob( G(0)) = { Z 2 ω } and Mor G(0)(Z 2 ω, Z 2 ω) is the set of all continuous group homomorphisms from Z 2 ω into Z 2 ω.